PERT DISTRIBUTION
ABSTRACT
The PERT distribution is one of the most popular probability continuous distributions with applications to real life data. In this paper some structural properties of this distribution such as Moments, moment generating function, Characteristics function, Cumulative distribution function, Survival function, Hazard function, and also derive the Information matrix. The plots of the Probability density function, cumulative distribution function, Survival function and hazard function of the PERT distribution were constructed for ease of understanding of its shapes under different parameter combinations. Keywords: PERT Distribution, Moment Generating Function, Survival Function, Hazard Function, Information Matrix.
INTRODUCTION
The PERT distribution (also known as the Beta-PERT distribution gets its name because it uses the same assumption about the mean (see below) as PERT networks (used in the past for project planning). It is a version of the Beta distribution and requires the same three parameters as the Triangle distribution, namely minimum (a), mode (b) and maximum (c). The figure below shows three PERT distributions whose shape can be compared to triangle distribution:
Uses
The PERT distribution is used exclusively for where one is given the expert’s minimum, most likely and maximum guesses. It is a direct alternative to a Triangle distrubution so a discussion is warranted on comparing the two:
Comparison with the Triangle distribution
The equation of a PERT distribution is related to the Beta4 distribution as follows:
PERT (a, b, c) = Beta4(a1, a2, a, c)
where:
The last equation for the mean is a restriction that is assumed in order to be able to determine values for a1 and a2. It also shows how the mean for the PERT distribution is four times more sensitive to the most likely value than to the minimum and maximum values.
This should be compared with the Triangle distribution where the mean is equally sensitive to each parameter. The PERT distribution therefore does not suffer to the same extent the potential systematic bias problems of the Triangle distribution, that is in producing too great a value for the mean of the risk analysis results where the maximum for the distribution is very large.
The standard deviation of a PERT distribution is also less sensitive to the estimate of the extremes. Although the equation for the PERT standard deviation is rather complex, the point can be illustrated very well graphically. The figure below compares the standard deviations of the Triangle and PERT distributions with minimum a=0, maximum c= 1, and varying most likely value b.
The observed pattern extends to any {a,b,c} set of values. The graph shows that the PERT distribution produces a systematically lower standard deviation than the Triangle distribution, particularly where the distribution is highly skewed (i.e. b is close to the minimum or maximum). As a general rough rule of thumb, cost and duration distributions for project tasks often have a ratio of about 2:1 between the (maximum - most likely) and (most likely - minimum), equivalent to b = 0.3333 on the figure above. The standard deviation of the PERT distribution at this point is about 88% of that for the Triangle distribution. This implies that using PERT distributions throughout a cost or schedule model, or any other additive model with similar ratios, will display about 10% less uncertainty than the equivalent model using Triangle distributions.
You might argue that the increased uncertainty that occurs with Triangle distributions will compensate to some degree for the over confidence that is often apparent in subjective estimating. The argument is quite appealing at first sight but is not conducive to the long term improvement of the organization’s ability to estimate. We would rather see an expert’s opinion modelled as precisely as is practical. Then, if the expert is consistently over-confident, this will become apparent with time and his/her estimating can be re-calibrated.
Limitations to using the PERT distribution
The PERT distribution came out of the need to describe the uncertainty in tasks during the development of the Polaris missile . The project had thousands of tasks and estimates needed to be made that were intuitive, quick and consistent in approach. The Four-Parameter Beta distribution was used just because it came to the author’s mind (the Kumaraswamy distribution would also have been a good candidate, for example). The decision to constrain the distribution so that it’s Mean = (Min + 4* Mode + Max)/6 was an approximation to their decision that the distribution should have a standard deviation of 1/6 of its range (i.e. Max - Min). demonstrated that, if one wishes to maintain this [standard deviation = range/6] idea then the PERT distribution should only be used with a certain range of values for the mode, namely:
Mode + 0.13(Max - Mode) < Mode < Mode + 0.13(Max - Mode)
i.e. that the mode should not lie less that 13% of the range from either the Min or Max values. In practice this is pretty good advice, and tends to occur when one has a very high Max value relative to the Min and Mode, since the distribution is very skewed and gives very small density in the extreme tail making the Max value estimate rather meaningless, for example:
describes a study that analyzed many PERT networks and concluded that “the”most likely” activity-time estimate m [mode] is practically useless”. They found that the location of the mode in most project tasks was approximately one third of the distance from the Min to the Max, i.e:
Mode = Min + (Max-Min)/3
Taking the Beta4(a1 ,a2,min, max) distribution again, this equates to a1 = 2, a2= 3. Thus, from Golenko-Ginzburg’s viewpoint it is sufficient to use
Beta4(2, 3, min, max)
in place of
PERT(min, mode, max)
with the added advantage that one is only asking a subject matter expert for two values.
FORMULA’S
APPLICATIONS
Probably the reason it gets so little respect, and in fact is so little known, is its humble origin in the trenches of project management. I first encountered it as a way to quantify uncertainty in the duration of tasks in project management. Hence the origin of the name – PERT is a project management technique made famous for its contributions to the US Navy's Polaris submarine program.
The analyst decides upon lower, most-likely, and upper values for some parameter, like the time to complete development on the frazzilated thingum module. Then weights of 1/6, 2/3, and 1/6 are assigned to these three to get a probability mass function for the parameter. Do this for all the uncertain durations in your project dependency (PERT) diagram, do the math, and you get a probability distribution for the completion time of the project, along with any milestones of interest. It's nice to have a computer to grind out the details.
This distribution has three parameters, which you can think of as the minimum, the most likely, and the maximum values, and a rule of 1/6 - 2/3 - 1/6 for the allocation of weights. Now this can be generalized in one or two ways. One is by changing how peaked the distribution is. Maybe the most-likely should have a weight of 9/10, or ½. This we shall call the "confidence" we have that the true value is near the most-likely. Another is by skewing the most-likely toward one end or the other. I haven't really said that the most-likely has to be in the middle, though that's what I've seen, so this is only a "maybe" generalization.
PROBLEM AND SOLUTION
For the given activities determine:
1. Critical path using PERT.
2. Calculate variance and standard deviation for each activity.
3. Calculate the probability of completing the project in 26 days.
| Activity | to | tm | tp |
|---|---|---|---|
| 1-2 | 6 | 9 | 12 |
| 1-3 | 3 | 4 | 11 |
| 2-4 | 2 | 5 | 14 |
| 3-4 | 4 | 6 | 8 |
| 3-5 | 1 | 1.5 | 5 |
| 2-6 | 5 | 6 | 7 |
| 4-6 | 7 | 8 | 15 |
| 5-6 | 1 | 2 | 3 |
Solution: First of all draw the network diagram for given data as shown below:
Here the time for completion of activities are probabilistic. So, using given values of time we will find the expected time to completion the activities and variance.
Expected time te=6to+4tm+tp
Variance V=(6tp−to)2
For each given activity we will calculate the expected time as follows:
| Activity | to | tm | tp | te=6to+4tm+tp | Variance (V) |
|---|---|---|---|---|---|
| 1-2 | 6 | 9 | 12 | 66+4×9+12=9 | (612−6)2=1.000 |
| 1-3 | 3 | 4 | 11 | 5 | 1.778 |
| 2-4 | 2 | 5 | 14 | 6 | 4.000 |
| 3-4 | 4 | 6 | 8 | 6 | 0.444 |
| 3-5 | 1 | 1.5 | 5 | 2 | 0.444 |
| 2-6 | 5 | 6 | 7 | 6 | 0.111 |
| 4-6 | 7 | 8 | 15 | 9 | 1.778 |
| 5-6 | 1 | 2 | 3 | 2 | 0.111 |
Now based on estimate time, we calculate the EST, EFT, LST and LFT for each activity to find out critical path of project as shown below.
| Activity | Duration | EST | EFT | LST | LFT | Total Float |
|---|---|---|---|---|---|---|
| 1-2 | 9 | 0 | 9 | 0 | 9 | 0 |
| 1-3 | 5 | 0 | 5 | 4 | 9 | 4 |
| 2-4 | 6 | 9 | 15 | 9 | 15 | 0 |
| 3-4 | 6 | 5 | 11 | 9 | 15 | 4 |
| 3-5 | 2 | 5 | 7 | 20 | 22 | 15 |
| 2-6 | 6 | 9 | 15 | 18 | 24 | 9 |
| 4-6 | 9 | 15 | 24 | 15 | 24 | 0 |
| 5-6 | 2 | 7 | 9 | 22 | 24 | 15 |
Here the critical path is along the activities 1-2, 2-4, 4-6. So the critical path is 1-2-4-6. Following diagram is prepared to show critical path along with EST and LFT.
∴ The critical path = 1-2-4-6 with time duration (tcp) of 24 days.
Here standard deviation is calculated for activities of critical path. So we get
σσ=V1+V2+V3=1+4+1.778=2.6034
Now the probability of completion of project in that given time (t) of 26 days, can be calculate by below formula,
Z=σt−tcp=2.603426−24=0.7682
Using table in Appendix-B, we get probability =77.8%
CONCLUSION
The PERT distribution produces a bell-shaped curve that is nearly normal. The PERT distribution with unknown end points was investigated as regard maximum likelihood estimation of its parameters. The maximum likelihood equations are derived along with the information matrix. With some assumptions, the information matrix I can be used to establish a minimum variance bound for an unbiased estimator by means of Cramer-Rao inequality. Also, under suitable regularity conditions, consistency and symptotic normality and efficiency can be claimed for the ML estimates. Then the diagonal elements of II are the asymptotic variances of the parameter estimates; however, they turn out to be quite a task to obtain in closed form. Consequently I was inverted numerically for specific values of a, c, b, α, and β.
REFERENCES
[1] Azaron, A., Katagiri, H., & Sakawa, M. (2007). Time-cost trade-off via optimal control theory in Markov PERT networks. Annals of Operations Research, 150, 47–64. [2] Ash, R.C. and Pittman, P.H., 2008. Towards holistic project scheduling using critical chain methodology enhanced with PERT buffering. International Journal of Project Organisation and Management 1(2), 185-203 [3] Ben- Yair A., Upon implementing the beta distribution in project management, Department of Industrial Engineering and Management, Negev academic college of Engineering, Israel. 2010. http://braude. ort. org. il/industrial/13th conf/html/%5 cfiles% 5c113-p. pdf.